Maass Forms in Algebra, Arithmetic Geometry, Combinatorics, Representation Theory, and String Theory

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Einstein's general relativity was formulated just over a hundred years ago, in 1915. Although there was no practical application available then, the theory now underpins the Global Positioning System, an almost ubiquitous feature of modern life. Just a year earlier, in 1914, the Indian mathematical genius and autodidact Ramanujan had arrived in England to work with Cambridge mathematician G. H. Hardy. The observations and insights he shared then continue to fascinate the mathematical world, and his personal story has captured the public imagination. In recent years evidence has emerged that the work of Einstein and the work of Ramanujan are related. This research project aims to develop the theory underlying this connection and lay important groundwork for future applications. The project is dedicated to the development of the theory of harmonic Maass forms and the application of this theory to topics in number theory, representation theory, and mathematical physics. Harmonic Maass forms generalize modular forms, and are now recognized as furnishing a theoretical framework for Ramanujan's mock theta functions. Freeman Dyson anticipated a role for the mock theta functions in string theory in 1987, and the currently-developing umbral moonshine theory is producing strong evidence in support of this vision. Building on their earlier work in these areas, the investigators aim to develop algebraic, analytic, and combinatorial tools for harmonic Maass forms that will shed light upon problems such as ranks of elliptic curves, the relationship between monstrous and umbral moonshine, and the consequences of this relationship for physics.